If you can use/install an anti-keylogger program, that means that someone else can install a rootkit as well. You can try to install or startup a virtual keyboard or some other tool and see if that works. If so, the machine is probably not safe to start with. It's probably safer to use one that you can't mess with.
Take your own laptop, ipad, smartphone, and use wifi at a local bar. Use https-everywhere if possible, or set gmail, facebook etc to use https only. Use VPN if you're really worried.
NB: I can imagine that a machine is not safe and you can mess with it. Someone uses it and then the session is closed. If the machine is reset back to a clean snapshot after each closed session, it might be relatively safe, in the sense that if you stay away from unsafe sites, you'll probably stay clean. So if it appears to work like that, you can start a new session, browse facebook, hotmail, gmail, and stay safe. Then you start a new session, browse whatever unsafe site you want, don't login to gmail etc....